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NATIONAL SENIOR CERTIFICATE EXAMINATION NOVEMBER 2016

MATHEMATICS: PAPER II

EXAMINATION NUMBER

Time: 3 hours 150 marks

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of 27 pages and an Information Sheet of 2 pages (i–ii). Please

check that your paper is complete. 2. Read the questions carefully. 3. Answer ALL the questions on the question paper and hand this in at the end of the

examination. Remember to write your examination number on the space provided. 4. Number your answers exactly as the questions are numbered. 5. Diagrams are not necessarily drawn to scale. 6. You may use an approved non-programmable and non-graphical calculator, unless otherwise

stated. 7. Ensure that your calculator is in DEGREE mode. 8. Round off your answers to one decimal digit where necessary, unless otherwise stated. 9. All the necessary working details must be clearly shown. 10. It is in your own interest to write legibly and to present your work neatly.

FOR OFFICE USE ONLY: MARKER TO ENTER MARKS

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 TOTAL

8 24 11 16 8 8 12 10 16 11 9 17 /150

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NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27

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NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 SECTION A QUESTION 1 In the diagram below, straight line AB makes an angle of 135° with the y-axis. AB cuts the y-axis at A. B is joined to C, a point on the x-axis, so that BC is parallel to the y-axis.

(a) Explain why ABCO is not a cyclic quadrilateral.

(1) (b) If OA 8= units determine the equation of the line AB.

(3)


NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 (c) If OC 6= units then determine

(1) the equation of BC.

(1)

(2) the area of OCBA.

(3) [8]


NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 QUESTION 2

If 22sin (180 ) sin 2M

cos 2θ θθ

° − += and 2sinP

cos sinθ

θ θ=

−

(a) (1) Prove that M P.=

(5)



(2) For what value(s) of θ will P be undefined if [ ]180 ;360 ?θ ∈ − ° °

(5)



(b) If 112sin =β and cos 0β < then

(1) In which quadrant is angle β? (Circle the correct number) (1)

(2) Without the use of a calculator, determine the value of tan .β

(4)

12

43


NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 (c) (1) Prove that ( ) ( )cos 30 cos 30 sin .α α α− ° − + ° =

(4)

(2) Hence, determine the general solution to the equation:

2cos( 30 ) cos( 30 ) 2sin .α α α− ° − + ° =

(5)

[24]


NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 QUESTION 3 In the diagram below: • Circle P touches the x-axis and the y-axis. • Circle P touches circle Q at one point. • Circle Q touches circle R at one point. • Circle P has a radius of 5 units. • PQ is parallel to the x-axis and RQ is parallel to the y-axis. • PR cuts the circles with centres P and R at A and B respectively.

(a) If PQ 9= units, determine the equation of circle centre Q.

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NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 (b) Find the length of line QR if the equation of circle centre R is 2 2( ) 22 117.x p y y− + − = −

(3) (c) Determine the length of AB, correct to two decimal digits.

(4) [11]


NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 QUESTION 4 (a) Use the diagram below to prove the theorem that states that the opposite angles of a

cyclic quadrilateral are supplementary.

(5)


NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 (b) In the diagram below:

• O is the centre of the circle. • A, B and C lie on the circumference of the circle. • Line DAE is a tangent to the circle at point A. • 4A 62 .= ° • 2A 25 .= °

Calculate the size of angle 1C .

(6)


NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 (c) (1) What single statement can be deduced from both of the following

statements?

M + N = D M + Q = D

(1)

(2) In the diagram below:

• A, B, C and D lie on the circumference of the circle.

Prove that 1 2ˆ ˆB A C .= +

(4) [16]

A

D

B

C

12

1

2

1

2


NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 QUESTION 5 The graph of [ ]( ) 3sin 1 with 0 ;360f x x x= + ∈ ° ° is drawn below. Line AB is parallel to the x-axis. A is a point on the y-axis.

(a) Write down the coordinates of point B.

(2)

(b) Calculate the value(s) for x where ( ) 1f x = − if [ ]0 ;360 .x∈ ° °

(4)

(c) If ( )g x k= and k is a constant term, then for what value(s) of k will ( ) ( )f x g x=

have no real solutions if [ ]0 ;180 ?x∈ ° °

(2)

[8]


NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 QUESTION 6 The manager of a hardware store records the number of staff that are in the store on a given day as well as the sales generated that day.

Staff 20 5 17 7 4 8 15 1 10 12 23 14

Sales in rand 35 200 9 200 32 000 15 600 9 200 17 200 31 200 3 000 19 600 26 800 39 200 20 800

(a) Calculate the correlation coefficient for the data above and then comment on the

strength of the relationship.

(3) (b) Find the equation for the line of best fit in the form ...y =

(2) (c) The manager decides to use 19 staff in his store. At the end of the day the sales is

R23 000. Would the manager consider this a successful day? Explain your answer.

(3) [8] Total for Section A: 75 marks


NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 SECTION B QUESTION 7 You develop a product and do some market research. The table below is a summary of the ages of people who say they will buy the product.

Age Mid-point Frequency Cumulative Frequency

5 15x< ≤ 10 200 200 15 25x< ≤ 20 A 450 25 35x< ≤ 30 20 470 35 45x< ≤ 40 32 B 45 55x< ≤ 50 23 525 55 65x< ≤ 60 300 825 65 75x< ≤ 70 475 1300

(a) Calculate the values of A and B in the table above.

(2) (b) (1) Calculate the estimated mean age of the people who say they will buy the

product.

(2) (2) Find the modal class interval.

(1)


NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 (c) Sketch the Ogive on the grid below.

(3) (d) (1) Is the data normally distributed? Explain.

(2)

(2) You are developing a marketing strategy. Is the mean age a good indicator

of how to advertise your product? Explain your answer.

(2) [12] IEB Copyright © 2016 PLEASE TURN OVER

NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 QUESTION 8 In the diagram below, circle centre T touches the x-axis at R. • AO is a tangent to the circle at P. • OT 5= and TP 3.=

(a) Determine the coordinates of T.

(5)



(b) Determine ˆTOR.

(2)

(c) Determine ,Py the y-coordinate of point P.

(3)

[10]


NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 QUESTION 9 In the diagram below, C and A are points on the x-axis and y-axis respectively. • OB AC.⊥ • OB has a length of 20 units. • BC has a length of 80 units.

(a) Determine the length of OC.

(2)



(d) Prove that ABO / / / OBC∆ ∆ and hence deduce that 2OBAB .

BC=

(5)

[16]


NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 QUESTION 10 The diagram below is the top view design of a new railway system. There are eight stations being built and these are labelled with letters from A–H. You have been asked to do some calculations for the railway company. As the engineer you know that: • AF//BE and AC//GD.

• AB 4BC 7

= and AG 9 .AF 17

=

(a) Calculate FE .FC

(3)

(b) Calculate CD .DF

(2)

A

B

CDEF

G

H


NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 (c) If the straight line distance of the track from F to C is 374 kilometres and it takes

50 hours to build one kilometre of the track, determine the number of hours it will take to build the section from E to D.

(6)

[11]


NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 QUESTION 11 In the diagram below: • B, C, D, E and F lie on the circle centre O. • Lines AB and AF are tangents to the circle at B and F respectively. • Line BE passes through O.

(a) Prove that 1ˆ ˆC D 90 .+ = °

(4)

(b) If D 38 ,= ° determine the size of ˆBAF.

(5) [9]


NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 QUESTION 12 In the diagram below: • A, B, C and D lie on the circle centre O. • P is a point on BC so that OP BC.⊥ • AD DC CP 6 units.= = = • ˆˆADC 130 and BCD .θ= ° =

(a) Determine the area of ∆ADC.

(2) (b) Show that ˆDBC 25 .= °

(4)


NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II of 27 (c) Calculate the value of θ if θ 90 .< °

(6)

(d) Given that A, D and C remain fixed points on the circle and that point B is lifted off the plane containing the circle and positioned at a point, T, which is 9 units vertically above point A. Determine ˆTCA.

(5)

[17] Total for Section B: 75 marks Total: 150 marks


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